2011/09/02 by Árpád Bényi, Branko Ćurgus, Bényi, Árpád +1
Mathematics · #15A24 #15B05 #20H15 #51F15 #51M04 #51M15 #51N20 #FOS: Mathematics #History and Theory of Mathematics #Mathematical Dynamics and Fractals #Mathematics and Applications #Metric Geometry (math.MG) #math.MG #msc:15A24 #msc:15B05 #msc:20H15 #msc:51F15 #msc:51M04 #msc:51M15 #msc:51N20
paper · pdf · doi:10.48550/arxiv.1109.0557
33 pages, 17 figures
openalex publication_date 2011/09/02 · arxiv created 2013/01/16 · arxiv updated 2013/01/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
For a given triangle T and a real number ρ we define Ceva's triangle \CTρ(T) to be the triangle formed by three cevians each joining a vertex of T to the point which divides the opposite side in the ratio ρ:(1-ρ). We identify the smallest interval \nMT ⊂ \nR such that the family \CTρ(T), ρ∈ \nMT, contains all Ceva's triangles up to similarity. We prove that the composition of operators \CTρ, ρ∈ \nR, acting on triangles is governed by a certain group structure on \nR. We use this structure to prove that two triangles have the same Brocard angle if and only if a congruent copy of one of them can be recovered by sufficiently many iterations of two operators \CTρ and \CTξ acting on the other triangle.